Mavko G., Mukerji T., Dvorkin J. The Rock Physics Handbook

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Then, the linear form of Gassmann’s relations can be approximated as

K

Gass

ðÞ



K

P-wave modulus form

Because Gassmann’s relations predict no change in the shear modulus, we can also

write the linear form of Gassmann’s relation as

M

Gass

ðÞ¼K

Gass

ðÞ



K

where DM

Gass

(f) is the predicted change in the P-wave modulus M ¼ K þ

Similarly, we can write

K

Gass

¼ M

Gass

¼ l

Gass

where l is Lame

’s parameter.

Velocity form

Murphy et al. (1991) suggested a velocity form of Gassmann’s relation



sat

Psat

¼K

þ K

dry

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

Porosity

0.4

0.3

0.2

0.5

A⬘

Reuss (water)

Reuss (air)

Wate

Air

= 0.1

K/K

(f = 1 only)

Figure 6.3.1 f

is the porosity where a straight line drawn from the mineral modulus on the

left-hand axis, through the data point (A or A

), intersects the Reuss average for that corresponding

pore fluid. The change in rock bulk modulus between any two pore fluids (A–A

) is proportional to

the change in Reuss average for the same two fluids evaluated at the intercept porosity f

277 6.3 Gassmann’s relations: isotropic form



sat

Ssat

¼ m

which is easily written as

Psat

Ssat

dry

where r

sat

is the density of the saturated rock, V

Psat

is the P-wave saturated rock

velocity, V

Ssat

is the S-wave saturated rock velocity, K

dry

is the dry-rock bulk modulus,

m ¼m

dry

¼m

sat

is the rock shear modulus, and

1  K

dry



=K

þð1  Þ=K

 K

dry

Pore stiffness interpretation

One can write the dry-rock compressibility at constant pore pressure, K

1

dry

, as (Walsh,

1965; Zimmerman, 1991a)

dry



where



]



is the effective dry-rock pore-space compressibility, defined as the ratio of the

fractional change in pore volume, v

, to an increment of applied external hydrostatic

stress, s,atconstant pore pressure (see Section 2.9). This is related to another

pore compressibility, K

1

P

, which is more familiar to reservoir engineers and

hydrogeologists:

 P

¼





which is the ratio of the fractional change in pore volume to an increment of applied

pore pressure, at constant confining pressure, also expressed by (Zimmerman,

1991a)

P





Figure 6.3.2 shows a plot of normalized dry bulk modulus, K

dry

, versus porosity,

computed for various values of normalized pore-space stiffness, K

278 Fluid effects on wave propagation

The data points plotted in Figure 6.3.2 are dynamic bulk moduli (calculated

from ultrasonic velocities) for: (1) ten clean sandstones, all at a pressure of 40 MPa

(Han, 1986); (2) a single clean sandstone at pressures ranging from 5 to 40 MPa

(Han, 1986); and (3) porous glass (Walsh et al., 1965). Effective pore-space com-

pressibilities for each point can be read directly from the contours.

Similarly, the low-frequency saturated rock compressibility, K

1

sat

, can be written as

sat



where



¼ K



 K

 K



þ K

Here K

is the pore-fluid bulk modulus and K

is the same dry pore-space stiffness

defined above. The functional form for K

sat

is exactly the same as for K

dry

. The

difference is only in the term



, which is equal to the dry-pore stiffness, K

incremented by a fluid term F ¼(K

)/(K

– K

)  K

. Hence, the only effect of

a change in fluid is a change in the modified pore stiffness



Therefore, fluid substitution, as predicted by Gassmann’s equation, can be applied

by computing this change, 



, and then jumping the appropriate number of

contours, as illustrated in Figure 6.3.3.

For the example shown, the starting point A was one of Han’s (1986) data points

for an effective dry-rock bulk modulus, K

dry

¼0.44, and porosity, f ¼0.20.

Because the rock is dry, F

(1)

 0. To saturate with water (K

 0.056), we

0.2

0.4

0.6

0.8

0.2

0.4

0.6

0.8

Porosity

0.4

0.3

0.2

0.5

dry

Modified Voigt

Voigt

Reuss

Porous glass

ss - high P

ss - variable P

Kf/K

= 0.1

0 0.2 0.4 0.6 0.8 1

Figure 6.3.2 Normalized dry bulk modulus, K

dry

, versus porosity, computed for various

values of normalized pore-space stiffness, K

279 6.3 Gassmann’s relations: isotropic form

move up the amount (F

(2)

– F

(1)

)/K

¼0.06, or three contours. The water-saturated

modulus can be read off directly as K

sat

¼0.52, point A

. Obviously, the technique

works equally well for going from dry to saturated, saturated to dry, or from one fluid

to another. All we do is compute DF/K

and count the contours.

The second example shown (points B–B

) is for the same two pore fluids and the same

porosity. However, the change in rock stiffness during fluid substitution is much larger.

This illustrates the important point that a softer rock will have a larger sensitivity to

fluid substitution than a stiffer rock at the same porosity. In fact, anything that causes

the velocity input to Gassmann’s equation to be higher (such as a stiffer rock, a

measurement error, or velocities contaminated by velocity dispersion) will result in a

smaller predicted sensitivity to changes in pore fluids. Similarly, lower input veloci-

ties will lead to a larger predicted sensitivity.

Biot coefficient

The dry-rock modulus can be written as

dry

¼ K

ð1  b Þ

where K

dry

and K

are the bulk moduli of the dry rock and the mineral, and b is

sometimes called the Biot coefficient, defined as the ratio of pore-volume change Dv

to bulk-volume change, D V, at constant pore pressure:

b ¼

v

V



dry

K

dry



¼ 1 

dry

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

Porosity

0.4

0.3

0.2

0.5

K/K

(f = 1 only)

⬘

B⬘

f/Kf = 0.1

Figure 6.3.3 Normalized low-frequency saturated bulk modulus, K/K

, versus porosity,

computed for various values of the normalized modified pore-space stiffness,



280 Fluid effects on wave propagation

Then, Gassmann’s equation can be expressed as

sat

¼ K

dry

þ b

where

b  



but no V

In practice, fluid substitution is performed by starting with compressional- and shear-

wave velocities measured on rocks saturated with the initial pore fluid (or gas) and

then extracting the bulk and shear moduli. Then the bulk modulus of the rock

saturated with the new pore fluid is calculated by using Gassmann’s relation, and

the velocities are reconstructed.

A practical problem arises when we wish to estimate the change of V

during fluid

substitution, but the shear velocity is unknown, which is often the case in situ. Then,

strictly speaking, K cannot be extracted from V

, and Gassmann’s relations cannot be

applied. To get around this problem, a common approach is to estimate V

from an

empirical V

–V

relation or to assume a dry-rock Poisson ratio (Castagna et al., 1985;

Greenberg and Castagna, 1992; see Section 7.9).

Mavko et al. (1995) have suggested an approximate method that operates directly

on the P-wave modulus, M ¼ V

. The method is equivalent to replacing the bulk

moduli of the rock and mineral in Gassmann’s relation with the corresponding

P-wave moduli. For example,

sat

 M

sat



dry

 M

dry

ðM

 M

where M

sat

, M

dry

, M

, and M

are the P-wave moduli of the saturated rock, the dry

rock, the mineral, and the pore fluid, respectively. The approximate method performs

the same operation with the P-wave modulus, M, as is performed with the bulk

modulus, K, in any of the various exact forms of Gassmann’s relations listed above.

Uses

Gassmann’s relations are used to estimate the change of low-frequency elastic moduli

of porous media caused by a change of pore fluids.

Assumptions and limitations

The following considerations apply to the use of Gassmann’s relations.



low seismic frequencies are assumed, so that pore pressures are equilibrated

throughout the pore space. In-situ seismic conditions should generally be acceptable.

281 6.3 Gassmann’s relations: isotropic form

Ultrasonic laboratory conditions will generally not be described well with

Gassmann’s equation. Sonic-logging frequencies may or may not be within the range

of validity, depending on the rock type and the fluid viscosity;



the rock is isotropic;



all minerals making up the rock have the same bulk and shear moduli; and



fluid-bearing rock is completely saturated.

Extensions

Gassmann’s relations can be extended in the following ways:



for mixed mineralogy, one can usually use an effective average modulus for K

;



for clay-filled rocks, it sometimes works best to consider the “soft” clay to be part

of the pore-filling phase rather than part of the mineral matrix. Then the pore fluid

is “mud,” and its modulus can be estimated with an isostress calculation, as in the

next item; and



for partially saturated rocks at sufficiently low frequencies, one can usually use an

effective modulus for the pore fluid that is an isostress average of the moduli of the

liquid and gaseous phases (see Section 6.17):

1  S

where K

is the bulk modulus of the liquid phase, K

is the bulk modulus of the gas

phase, and S is the saturation of the liquid phase.



For rocks with mixed mineralogy, use Brown and Korringa’s extension of Gassmann’s

relations (see Section 6.4).



For anisotropic rocks, use the anisotropic form of Gassmann’s relations (see

Section 6.5).



Berryman and Milton (1991) have extended Gassmann’s relation to include

multiple porous constituents (see Section 6.6).

6.4 Brown and Korringa’s generalized Gassmann equations

for mixed mineralogy

Synopsis

Brown and Korringa (1975) generalized Gassman’s (1951) isotropic fluid substitution

equations to the case where the solid phase of the rock is heterogeneous (i.e., mixed

mineralogy).

sat

 K

sat

dry

 K

dry

S

 K

S

 K



; m

sat

¼ m

dry

282 Fluid effects on wave propagation

where K

dry

is the dry-rock bulk modulus, K

sat

is the bulk modulus of the rock

saturated with a pore fluid, K

is the fluid bulk modulus, m

dry

is the dry-rock shear

modulus, m

sat

is the saturated-rock shear modulus, and f is porosity.

Two additional moduli K

and K

are defined as

¼





¼ constant

S

¼







¼ constant

where V is the total volume of a sample of rock, V

is the volume of pore space within

the sample, P

is the confining pressure, P

¼ P

 P

is the differential pressure,

and P

is the pore pressure. These special moduli (see also Zimmerman, 1991a)

describe the effect on sample volume and pore volume of equally incrementing

confining pressure and pore pressure – sometimes called the “unjacketed” case.

Caution

The notation here is different from that in Brown and Korringa (1975) and

Zimmerman (1991a).

The Brown and Korringa fluid substitution equation can also be written in terms

of compressibilities

sat

 C

ðÞ

1

¼ C

dry

 C



1

þ  C

 C

S



1

where C

sat

¼ 1=K

sat

; C

dry

¼ 1=K

dry

; C

¼ 1=K

, and C

S

¼ 1=K

S

When the mineral phase is homogeneous, then K

¼ K

S

¼ K

mineral

, and

¼ C

S

¼ C

mineral

, where C

mineral

¼ 1=K

mineral

, and K

mineral

is the mineral bulk

modulus. Thus, if there is only a single mineral, both the modulus and compress-

ibility forms of fluid substitution presented here reduce exactly to the isotropic

Gassmann equations.

A practical problem with using the Brown and Korringa equations for fluid

substitution is that we seldom have knowledge of C

and C

S

(or K

and K

S

). One

useful constraint comes from the relation

S





1  



mineral

where C

mineral

¼ 1=K

mineral

is the true effective compressibility of the mineral mix.

mineral

must satisfy the Hashin–Shtrikman bounds, K

HS

mineral

, and might be estimated,

for example, using a Voigt–Reuss–Hill average of the individual mineral bulk

moduli. Insights into the value of K

come from knowledge of the grain

283 6.4 Brown and Korringa’s generalized Gassmann equations

microstructure. K

might lie close to the upper bound of the mineral bulk modulus,

if the softest phases, such as clay, are pore-filling. On the other hand, if the softest

minerals are load-bearing, then K

might be closer to the lower bound. Once an estimate

of K

is made, a consistent estimate of Kf

can be found from the above relation.

Uses

The Gassmann and Brown–Korringa relations are used to estimate the change of low-

frequency elastic moduli of porous media caused by a change of pore fluids.

Assumptions and limitations

The following considerations apply to the use of Gassmann and Brown–Korringa

relations.



Low seismic frequencies are assumed, so that pore pressures are equilibrated

throughout the pore space. In-situ seismic conditions should generally be accept-

able. Ultrasonic laboratory conditions will generally not be described well with

Gassmann or Brown–Korringa equations. Sonic-logging frequencies may or may

not be within the range of validity, depending on the rock type and fluid viscosity.



The rock is isotropic.



Fluid-bearing rock is completely saturated.

6.5 Fluid substitution in anisotropic rocks

Synopsis

In addition to his famous isotropic equations, Gassmann (1951) published fluid-

substitution equations for anisotropic porous rocks. In terms of the anisotropic linear

elastic stiffness components, c

ijkl

, his result is written as

sat

ijkl

¼ c

dry

ijkl

 c

dry

ijaa



 c

dry

bbkl



ðÞ

 K

 K

ðÞ

þ K

 c

dry

ccdd



where c

dry

ijkl

is the effective elastic stiffness element of dry rock, c

sat

ijkl

is the effective

elastic stiffness element of rock saturated with pore fluid, K

is the mineral bulk

modulus, K

is the fluid bulk modulus, and f is porosity:

1 for i ¼ j

0 for i 6¼ j



The inverse of Gassmann’s anisotropic equation (not given in Gassmann’s paper) can

be written as

284 Fluid effects on wave propagation

dry

ijkl

¼ c

sat

ijkl



 c

sat

ijaa



 c

sat

bbkl



ðÞ K

 K

ðÞK

 c

sat

ccdd



Successive application of the second equation followed by the first one allows us to

apply fluid substitution from any initial fluid to any final fluid.

Similarly, Gassmann (1951) gave anisotropic fluid-substitution expressions in

terms of the elastic compliances:

sat

ijkl

¼ s

dry

ijkl



dry

ijaa

 d

=3K



dry

bbkl

 d

=3K



dry

ccdd

 1=K



þ  1=K

 1=K

ðÞ

where s

dry

ijkl

is the effective elastic compliance element of dry rock, s

sat

ijkl

is the effective

elastic compliance element of rock saturated with pore fluid, K

is the bulk modulus

of the solid mineral, K

is the bulk modulus of the fluid, and f is porosity.

The inverse expression (not in Gassmann’s original paper) is written as

dry

ijkl

¼ s

sat

ijkl

sat

ijaa

 d

=3K



sat

bbkl

 d

=3K



sat

ccdd

 1=K



  1=K

 1=K

ðÞ

Gassmann’s equations assume that the mineral is homogeneous and isotropic,

although the dry and saturated rocks can have arbitrary anisotropy. In all of the

equations of this section, a repeated index within a term implies a sum over 1–3 (e.g.,

ijaa

¼ c

ij11

þ c

ij22

þ c

ij33

Brown and Korringa (1975) also gave expressions for fluid substitution in aniso-

tropic rocks, expressed in the compliance domain. Their result is very similar to

Gassmann’s, except that they allow for an anisotropic mineral. (Brown and Korringa

also made a generalization to allow for mixed mineralogy. That generalization is shown

in Section 6.4 for isotropic rocks.)

sat

ijkl

¼ s

dry

ijkl



dry

ijaa

 s

ijaa



dry

bbkl

 s

bbkl



dry

ccdd

 s

ccdd



þ  b

 b

ðÞ

where s

dry

ijkl

is the effective elastic compliance tensor element of dry rock, s

sat

ijkl

the effective elastic compliance element of rock saturated with pore fluid, s

ijkl

the effective elastic compliance element of the solid mineral, b

is the fluid

compressibility ¼1=K

, b

is the mineral compressibility ¼s

aagg

¼ 1=K

, and f is

porosity.

The inverse is given by

dry

ijkl

¼ s

sat

ijkl

sat

ijaa

 s

ijaa



sat

bbkl

 s

bbkl



sat

ccdd

 s

ccdd



  1=K

 1=K

ðÞ

285 6.5 Fluid substitution in anisotropic rocks

The practical differences between the Gassmann and Brown–Korringa equations

presented here appear to be inconsequential. Even though mineral grains are usually

highly anisotropic, Brown and Korringa’s equations imply that all grains in the rock

are crystallographically aligned, which will not be the case in real rocks. Furthermore,

complete elastic tensors for rock-forming minerals are generally not well known. In

most cases, our only practical choice will be to assume an average isotropic mineral

modulus.

One of the major disadvantages in the application of the anisotropic Gassmann

equation is that in the field we seldom measure enough parameters to completely

characterize the stiffness tensor of a rock. Mavko and Bandyopadhyay (2008) find an

approximate form of fluid substitution, for vertically propagating seismic waves in a

transversely isotropic medium with a vertical axis of symmetry (VTI). The aniso-

tropic correction for the fluid effect depends only on Thomsen’s parameter d:

sat

3333

c

dry

3333

ðÞK

 K

dry

iso



 K

 K

ðÞþK

ðÞK

 K

dry

iso



1  d

dry

3333

 K

dry

iso

dry

3333

c

sat

3333



ðÞK

 K

sat

iso



 K

 K

ðÞK

ðÞK

 K

sat

iso



1  d

sat

3333

 K

sat

iso



where K

dry

iso

and K

sat

iso

are the apparent isotropic bulk moduli of the dry and saturated

rock, respectively, computed from the vertical P- and S-wave velocities:

iso

¼  V





. In these equations, the term within the square brackets is just

the difference between the dry and saturated apparent bulk moduli as predicted by the

isotropic Gassmann equation. The approximation is good for porosities larger than

the magnitude of the Thomsen parameters e; g; dðÞ.

Uses

The equations of Gassmann and Brown and Korringa are applicable to the fluid

substitution problem in anisotropic rocks.

Assumptions and limitations

The following considerations apply to the use of Brown and Korringa’s relations:



low seismic frequencies (equilibrated pore pressures). In-situ seismic frequencies

generally should be acceptable. Ultrasonic laboratory conditions will generally not

be described well with Brown and Korringa’s equations. Sonic-logging frequencies

may or may not be within the range of validity, depending on the rock type and

fluid viscosity;



all minerals making up the rock have the same moduli; and



fluid-bearing rock is completely saturated.

286 Fluid effects on wave propagation